How would I cut across a plane using a die-like mesh/curve?
To cut the shape on the plane with a curve object you can use Knife Project tool. Place your curve object above the plane, while its selected hold Shift and select the plane, switch to Top View, then Orthogonal View by hitting 5 on the Numeric Keypad and change to Edit Mode.  

Then In Edit Mode select Knife Project in Tools tab.

To avoid errors in the mesh, before using Knife Project tool you can subdivide the plane few times or convert the curve object to mesh.